Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations

Bounds on the growth of high Sobolev norms of

solutions to nonlinear Schrodinger equations

by

Vedran Sohinger

B.A., University of California, Berkeley (2006)

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

April 2011

@

Massachusetts Institute of Technology

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Abby Rockefeller

-ssACH6P U1 F-T-T INSTIT UTE OF TY

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Department of Mathematics

April 6, 2011

. ...

Gigliola Staffilani

Mauz6 Professor of Mathematics

Thesis Supervisor

Accepted by ...

/

Bjorn Poonen

Chairman, Department Committee on Graduate Theses

...

Bounds on the growth of high Sobolev norms of solutions to

nonlinear Schrddinger equations

by

Vedran Sohinger

Submitted to the Department of Mathematics on April 25, 2011, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

In this thesis, we study the growth of Sobolev norms of global solutions of solutions to nonlinear Schr6dinger type equations which we can't bound from above by energy conservation. The growth of such norms gives a quantitative estimate on the low-to-high frequency cascade which can occur due to the nonlinear evolution. In our work, we present two possible frequency decomposition methods which allow us to obtain polynomial bounds on the high Sobolev norms of the solutions to the equations we are considering. The first method is a high regularity version of the I-method previously used by Colliander, Keel, Staffilani, Takaoka, and Tao and it allows us to treat a wide range of equations, including the power type NLS equation and the Hartree equation with sufficiently regular convolution potential, as well as the Gross-Pitaevskii equation for dipolar quantum gases in the physically relevant 3D setting. The other method is based on a rough cut-off in frequency and it allows us to bound the growth of fractional Sobolev norms of the completely integrable defocusing cubic NLS on the real line.

Thesis Supervisor: Gigliola Staffilani

Acknowledgments

I would like to thank my Advisor, Gigliola Staffilani, for her dedication and

patience. Her determined guidance and vision made this work possible. It was an honor to be her student.

Over the years, I have had the privilege to study with great and inspiring pro-fessors, Damir Bakic, Boris Guljas, and Andrej Dujella at the University of Zagreb, Marina Ratner, Vaughan Jones, Maciej Zworski, Michael Christ, Lawrence Craig Evans, Michael Klass, William Kahan, and Olga Holtz at the University of Califor-nia Berkeley, and Richard Melrose, David Jerison, Tobias Colding, Tomasz Mrowka, and Denis Auroux at the Massachusetts Institute of Technology. I would also like to specially mention Zeljko Buranji, my math teacher at the Ivan Goran Kovacic Ele-mentary School in Zagreb, Croatia. To all of the mentioned teachers, I express my deepest gratitude. It was their motivation which drove me to seek to understand the beauty of mathematics.

I would like to express my gratitude to Antti Knowles for teaching a useful class

on dynamics of large quantum systems, as well as for suggesting several problems for our research, most notably the Hartree equation. Also, I thank Kay Kirkpatrick for suggesting to us to study the Gross-Pitaevskii equation for dipolar quantum gases. Moreover, I would like to thank Hans Christianson for many patient discussions about well-posedness theory and for his useful suggestions in our research, as well as for his kindness, encouragement, and sense of humor.

It was a great privilege to be a teaching assistant for the graduate analysis class

18.155 in the Fall of 2010, under the superb guidance of Michael Eichmair. Being a TA for the class was an excellent learning experience and I would like to thank both

Michael and the whole class for making it possible. Michael's wisdom and generosity have always been an inspiration to the whole department.

I am grateful for the support and friendship I have received over the last five years

from my friends Hadi Tavakoli Nia, Francesco Mazzini, Martina Balagovic, Carla Sofia Perez Martinez, Silvia Sabatini, Silvia Montarani, Matjaz Konvalinka, Tova Brown,

Martin Frankland, David Jordan, Natasa Blitvic, Tea Zakula, and Grgur Tokic. One can just wish to have such great friends.

During my graduate school years, I always enjoyed the friendship of my class-mates, colleagues, and friends at MIT, and I would like to thank Amanda Redlich, Alejandro Morales, Nikola Kamburov, Angelica Osorno, Ana Rita Pires, Vera Vertesi, Hoda Bidkhori, Niels Martin Moller, Lu Wang, Hamid Hezari, Xuwen Zhu, Michael Donovan, Dana Mendelson, Michael Andrews, Fucheng Tan, Rosalie Belanger-Rioux, Chris Dodd, Uhirinn Suh, Yoonsuk Hyun, Benjamin Iriarte Giraldo, Jennifer Park, Tiankai Liu, Kestutis Cesnavicius, Ramis Movassagh, Kiril Datchev, Hoeskulldur Haldorsson, Matthew Gelvin, Joel Lewis, Nate Bottman, Nan Li, Leonardo Andres Zepeda-Nunez, Bhairav Singh, Nikhil Savale, Dustin Clausen, Jethro van Ekeren, Ro-man Travkin, William Lopes, Ben Mares, Yan Zhang, Giorgia Fortuna, Liang Xiao, Jose Soto, Suho Oh, Christian Hilaire, Ailsa Keating, Jiayong Li, Hwanchul Yoo, Saul Glasman, Galyna Dobrovolska, Qian Lin, Giulia Sarfati, Roberto Svaldi, Ben Harris, Oleksandr Tsymbaliuk, Caterina Stoppato, Dorian Croitoru, Nicholas Sheridan, Nick Rozenblyum, Jacob Bernstein, Chris Kottke, Ronen Mukamel, Leonid Chindelevitch, Ricardo Andrade, Gregory Drugan, David Shirokoff, Chris Evans, Steven Sivek, Doris Dobi, Irida Altman, Khoa Lu Nguyen, Enno Lenzmann, Vera Mikyoung Hur, Shan-Yuan Ho, Christine Breiner, Emily Peters, Peter Tingley, Avshalom Manela, Brett Kotschwar, Gregg Musiker, Pierre Albin, James Pascaleff, Steven Kleene, Sam Wat-son, Aaron Naber, Chiara Toglia, Shani Sharif, Katarina Blagovic, Mohammad-Reza Alam, Ranko Sredojevic, Benjamin Charles Druecke, Alex Kalmikov, Wenting Xiao, Natasa Dragovic, Dusan Miljancevic, Mark Kalinich, Austin Minnich, Gunjan Agar-wal, Gerd Benjamin Bewersdorf, and Martin Kraus. They have helped make my MIT experience a pleasant one, and I will look back on these years with fondness.

I would like to thank my colleagues from conferences and workshops for their

helpful discussions. In particular, I would like to thank Jeremy Marzuola, Lydia Bieri, Robert Strain, Younghun Hong, Mahir Hadzic, Zaher Hani, Betsy Stovall, Naiara Arrizabalaga-Uriarte, Andoni Garcia-Alonso, Miren Zubeldia-Plazaola, Alessandro Selvitella, Paolo Antonelli, Ioan Bejenaru, Mihai Tohanaeanu, Boris Ettinger, and

Baoping Liu. I have learned a lot from discussions with my excellent colleagues. It was a great experience to work with superb classmates in college. I always fondly remember the study sessions with my classmates Ha Pham, Wenjing Zheng, Yann-Shin Aaron Chen, Boris Bukh, and Dominic McCarty at UC Berkeley. My classmates in college gave me great motivation to go to graduate school.

Furthermore, I would like to thank my friends in Croatia for believing in me and for their friendship despite my long absence. In particular, I would like to thank Marijan Polic, Daniela Sipalo, Marijana Zec, Ines Bonacic, Ana Prlic, Tomislav Beric, Dragana Pop, Ana Pavlina, Kresimir Ivancic, Matilda Troost, Rafaela Guberovic, and Elena Primorac. Their friendship has always meant a lot to me.

During the last four years, I have had the fortune to work with a superb piano teacher, Alice Wilkinson. I would like to thank her for her patience and her effort. Music has always been a balancing factor in my life, and I am grateful to my teacher

for helping me not to lose this important aspect.

I would like to especially thank my mother Jasminka Sohinger for her love and

ded-ication, and for being a superb life-long role model. In addition, I would like to thank my father Tomislav Sohinger for his constant and support and unsurpassable encour-agement. Finally, I would like to dedicate my thesis to the memory of my grandfather Ivan Sabo (1925-1994). His passion for knowledge and his open-mindedness have been an inspiration to me ever since I was very young.

Contents

1 Introduction 1.1 General setup

1.2 Statement of the problem . . . . 1.2.1 The NLS Cauchy problem . . . . 1.2.2 Conserved quantities . . . .

1.2.3 Global existence and a uniform bound . . . . . 1.2.4 Low-to-high frequency cascade . . . .

1.3 Previously known results . . . . 1.3.1 The linear Schr6dinger equation with potential . 1.4 Main ideas of our proofs . . . . 1.4.1 The smooth cut-off; the upside-down I-method . 1.4.2 The rough cut-off . . . .

1.5 Some notation and conventions . . . .

1.6 General facts from harmonic analysis . . . .

1.7 Organization of the Chapters . . . . 2 Bounds on S1

2.1 Introduction . . . .

2.1.1 Statement of the main results 2.1.2 Previously known results . . . 2.1.3 Main ideas of the proofs .

2.2 Facts from harmonic analysis . .

2.3 Quintic and Higher Order NLS .

13 . . . . 13 . . . . 14 . . . . 14 . . . . 15 . . . . 16 . . . . 16 . . . . 17 . . . . 20 . . . . 22 . . . . 23 . . . . 25 . . . . 26 . . . . 30 . . . . 35 37 . . . . 37 . . . . 39 . . . . 40 . . . . 4 1 . . . . 44 . . . . 45

2.3.1 Definition of the D operator . . . . 45

2.3.2 A local-in-time estimate and an approximation lemma . . . . 46

2.3.3 Control on the increment of |IDu(t)||

48

2.3.4 Proof of Theorem 2.1.1 for k > 3 . . . . 67

2.3.5 Remarks on the result of Bourgain . . . . 68

2.4 Modifications of the Cubic NLS . . . . 69

2.4.1 Modification 1: Hartree Equation . . . . 69

2.4.2 Modification 2: Defocusing Cubic NLS with a potential . . . . 88

2.4.3 Modification 3: Defocusing Cubic NLS with an inhomogeneous nonlinearity . . . . 101

2.4.4 Comments on (2.4), (2.5), and (2.6) . . . 105

2.5 Appendix A: Proof of Lemma 2.2.1 . . . 105

2.6 Appendix B: Proofs of Propositions 2.3.1, 2.3.2, and 2.3.3 . . . 109

3 Bounds on R 127 3.1 Introduction . . . 127

3.1.1 Statement of the main results . . . 127

3.1.2 _{Previously known results . . . .} 129

3.1.3 Main ideas of the proofs . . . . 130

3.2 Facts from harmonic analysis . . . 133

3.3 The cubic nonlinear Schr6dinger equation . . . 140

3.3.1 Basic facts about the equation . . . . 140

3.3.2 An Iteration bound and Proof of Theorem 3.1.1 . . . . 142

3.4 The Hartree equation . . . 152

3.4.1 Definition of the D operator . . . . 152

3.4.2 _{An Iteration bound and proof of Theorem 3.1.2 . . . . 154}

3.5 Appendix A: Auxiliary results for the cubic nonlinear Schradinger equation . . . 166

3.6 Appendix B: Auxiliary results for the Hartree equation . . . 169

4 Bounds on T2 and R2

4.1 Introduction . . . . 4.1.1 Statement of the main results . . . . 4.1.2 Previously known results . . . . 4.1.3 Main ideas of the proofs . . . . 4.2 Facts from harmonic analysis . . . . 4.2.1 Estimates on T2 . . . . 4.2.2 Estimates on R2 _{. . . .}

4.3 The Hartree equation on T2 . . . . 4.3.1 Definition of the D-operator . . . . 4.3.2 Local-in-time bounds . . . ..

4.3.3 A higher modified energy and an iteration bound

4.3.4 Further remarks on the equation . . . . 4.4 The Hartree equation on R . . . .

4.4.1 Definition of the D-operator . . . . 4.4.2 Local-in-time bounds . . . . 4.4.3 A higher modified energy and an iteration bound

4.4.4 Choice of the optimal parameters . . . . 4.4.5 Remarks on the scattering result of Dodson . . .

4.4.6 Further remarks on the equation . . . . 4.5 Appendix: Proof of Proposition 4.3.1 . . . .

5 Bounds on R3_{; the Gross-Pitaevskii Equation for dipo} gases

5.1 Introduction . . . . . . . .

5.1.1 Statement of the main result . . . .

5.1.2 Main ideas of the proof . . . .

5.2 Facts from harmonic analysis . . . . 5.2.1 An improved Strichartz estimate . . . . 5.2.2 A frequency localized Strichartz estimate

177 . . . . . . . 177 . . . . 178 . . . . 179 . . . . 180 . . . . 181 . . . . 181 . . . . 183 . . . . 187 . . . . 187 . . . . . 188 . . . . 190 . . . . 203 . . . . 204 . . . . 204 . . . . 205 . . . . 206 . . . . 218 . . . . 219 . . . . 221 . . . . 222 lar quantum 231 . . . . 231 . . . . 233 . . . . 233 . . . . 234 . . . . 235 . . . . 240

5.3 Proof of the Main Result . . . . 5.3.1 Definition of the E) operator . . . . .

5.3.2 Local-in-time bounds . . . .

5.3.3 Estimate on the growth of |IDu(t)||2 5.4 Appendix: Proof of Proposition 5.3.1 . . . . 5.5 Comments and further results . . . . 5.5.1 The unstable regime . . . .

5.5.2 Adding a potential . . . .

5.5.3 Higher modified energies . . . . 5.5.4 Lower dimensional results . . . .

. . . . 24 1 . . . . 24 1 . . . . 24 2 . . . . 24 3 . . . . 24 8 . . . . 258 . . . . 258 . . . . 258 . . . . 259 . . . . 26 1

Chapter 1

Introduction

1.1

General setup

Nonlinear Dispersive Partial Differential Equations model nonlinear wave phenomena which arise in various physical systems, such as the limiting dynamics of large Bose systems [92, 103], shallow water waves [79], and geometric optics [103]. These are non-linear evolution equations whose solutions spread out as waves in the spatial domain if no boundary conditions are imposed. The most famous examples of nonlinear dis-persive PDE are the Nonlinear Schrbdinger equation (NLS), the Korteweg-de Vries equation (KdV), and the Nonlinear wave equation (NLW). A key feature of these equations is that they are Hamiltonian, and hence they possess an energy functional which is formally conserved under their evolution.

The tools used to study nonlinear dispersive PDE come from harmonic analysis and from Fourier analysis. If one studies the equations on periodic domains, one also has to apply techniques from analytic number theory, as was first done in the work of Bourgain [9]. All of these tools are primarily used in order to understand the dis-persive properties of the linear part of the equation. These disdis-persive properties are manifested through an appropriate class of spacetime estimates known as Strichartz

estimates, as we will recall below. A family of Strichartz estimates in the non-periodic

re-solved by Keel-Tao [69]. In the work of Bourgain [9], Strichartz estimates were proved in the periodic setting. An appropriate use of Strichartz estimates and a fixed point ar-gument allows one to obtain local well-posedness in the critical or sub-critical regime'. In sub-critical regime, if one also has an a priori bound coming from conservation of an energy functional, one can easily obtain global well-posedness. Key contribu-tions to the study of the Cauchy problem for nonlinear dispersive PDE were made

by Ginibre-Velo [50, 51], Lin-Strauss [81], Kato [67], Hayashi-Nakamitsu-Tsutsumi [62], Cazenave-Weissler [29], Kenig-Ponce-Vega [72], and Bourgain [9]. These results

mostly concern the subcritical regime. There has also been a substantial amount of work done in the critical regime. Global existence results were proved in the

energy-critical regime, by Struwe [102], Grillakis [57, 58], Shatah-Struwe [94], Kapitanski

[66], Bahouri-Shatah [3], Bahouri-Gerard [2], Bourgain [16], Nakanishi [88, 89], Tao [105], Kenig-Merle [70], Ryckman-Visan [91], Visan [109],

Colliander-Keel-Staffilani-Takaoka-Tao [37], and in the mass-critical regime by Tao-Visan-Zhang [107], Killip-Visan-Zhang [76], Killip-Tao-Visan [74], Dodson [43, 44, 43].

Our work has focused on studying the qualitative properties of global solutions in the case of the NLS equation. The property that we want to understand is the transfer of energy from low to high frequencies. More precisely, we want to start out with initial data which are localized in frequency, and we want to see how fast a substantial part of the frequency support can flow to the high frequencies under the evolution of the NLS. As we will see below, one way to quantify this frequency cascade is through the growth in time of the high Sobolev norms of a solution u.

1.2

Statement of the problem

1.2.1

The NLS Cauchy problem

Given s > 1, we will consider the following general NLS Cauchy problem:

ist + Au = K(u),x c Xt E R

(1.1)

u(x, 0) = 4(x) E Hs(X).

Here, U = u(x, t), X is a spatial domain, t is the time variable, K is a nonlinearity, and HS(X) is the Sobolev space of index s on X. We will not study the general problem (1.1), but we will specialize to two possibilities in the spatial domain:

(i) X Td (Periodic setting). (ii) X Rd (Non-periodic setting).

We will restrict our attention to d ; 3. For the nonlinearity K(u), we will typically consider two main types:

(i) K(u) I uI2ku, for some k E N (Defocusing algebraic nonlinearity of degree

2k + 1).

(ii) K(u) = (V * |12)u, for some function V : X _{-+ R (Hartree nonlinearity).}

Here * denotes convolution in the x-variable:

f

fX

that the first nonlinearity is local, i.e its value at a point x depends only on the value of u at x, whereas the second nonlinearity is non-local. Some K(u) we will consider will be combinations and modifications of the algebraic and Hartree nonlinearities.

1.2.2

Conserved quantities

All the models we will consider will have conserved mass given by:

M(u(t)) =

Iu(x,t)|

and conserved energy given by:

E(u(t))

j

₊

j

_{P(u(x, t))dx.}

_(1.3)

The part P(u(x, t)) depends on the nonlinearity. We note that P(u) =-+ 2U|2k+2 for

the algebraic nonlinearity of degree 2k + 1 and P(u) = (V *

1u1

nonlinearity. The first term in E(u(t)) is called the kinetic energy and the second term is called the potential energy. In all of the models that we will be studying, the potential energy will be non-negative. We call this type of problem defocusing. One can also study the focusing in which the nonlinearity K(u) = -luI2k-1U. For such problems, energy is non longer necessarily a non-negative quantity. In this work, we will restrict our attention to the defocusing problem though.

The fact that mass and energy are conserved can be formally checked by differen-tiating under the integral sign. A rigorous justification requires a density argument which uses the well-posedness of the Cauchy problem. An overview of these ideas is given in [106].

1.2.3

Global existence and a uniform bound

Existence of solutions to (1.1) locally in time can be shown by using a fixed point argument [50, 81]. Since the initial data lies in H1, one can use the conservation of

mass and energy, as well as the fact that the potential energy is non-negative, one can deduce the existence of global solutions From the conservation of mass and energy and the non-negativity of the potential energy, it follows that the H' norm of a global solution is uniformly bounded in time [50, 51, 29, 9]. Namely, the following bound holds:

Ilu(t)IIHi

(1.4)

1.2.4

Low-to-high frequency cascade

We will be interested in obtaining bounds on the Hs norm of a solution. We recall that:

Here denotes the Fourier transform on X, and the domain of integration is Rd when

X = Rd and it is Zd when X = Td. If we combine Plancherel's Theorem and the

conservation of mass (1.2), it follows that the quantity:

(f

(

t)12d(9.

(1.6)

is constant in t. Hence, the area under the graph of the function -

|

the same for all times t. By (1.5) and (1.6), it follows that the growth of |Iu(t)||Hs for s

>>

|| >

cascade. We must note that it is not possible for the whole frequency support of u to

transfer to the high frequencies by (1.4). Hence, the growth of high Sobolev norms effectively estimates how much a only a part of the frequency support of a has moved to the high frequencies. We note that this problem sometimes also goes under the name of weak turbulence or wave turbulence and it has been studied since the 1960s

in the physics literature [60, 82, 117], and in the mathematical literature [6, 7]. The latter two papers were based on methods from probability theory. In the 1990s, the problem was also studied numerically [83]. The aim of all of the mentioned works is to obtain a statistical description of the forward cascade mechanism in weakly interacting dispersive wave models.

1.3

Previously known results

Suppose that u is a solution of (1.1). One can immediately obtain exponential bounds on the growth of Sobolev norms by iterating the local-in-time bounds coming from the local well-posedness of the equation. The main reason is that the increment time coming from local well-posedness is determined by the conserved quantities of the equation. More precisely, one recalls from [9, 15, 106] that there exist 6 > 0 and

||u(to + J)||H CUo)Hs 1.- )

We iterate (1.7) to obtain the exponential bound:

||u(t)|H s, <b, eAt (1.8)

It is, however, possible to obtain polynomial bounds. This was achieved for other nonlinear Schradinger equations in [12, 27, 98, 99, 118]. The main idea in these papers was to modify (1.7) to obtain an improved iteration bound by which there exists a constant r E (0, 1) depending on k, s and 6, C > 0 depending also on the initial data

such that for all times to:

||u(to + 6)|12

.

||(to)||2.

+

C||U(to)||1-,.

In [12], (1.9) bound is proved was in the use of approach, based in [27, 118].

is proved by the Fourier multiplier method, whereas in [98, 99], this

by using fine multilinear estimates. The key to the latter approach

smoothing estimates similar to those used in [73]. A slightly different on the analysis from [22], is used to obtain the same iteration bound

One can show that (1.9) implies:

||u(t)||Hs s4,, (1 + t|)1.1 (1.10)

A slightly different approach to bounding |Iu(t)IIH- is given in [13]. In this work, one considers the defocusing cubic NLS on R3 and by an appropriate use of Strichartz estimates, it is shown that, given 4 E H1(R3), one obtains the following uniform

bound on the localized Sobolev norms for the solution u:

||u(t)||H ,",,(R3) < C(4) (1.11)

Here If||H",, (R3) := SupICR3,Iis a unit cube 11fIH (I, where

||

corresponding restriction norm. However, the local bound (1.11) can be improved to a global one by using the results from [35] below, as we will see.

For certain NLS equations, one can deduce uniform bounds on

|ju(t)IHs

(1.1) is said to scatter in H5 if, for all <b c HS, there exist ua E H' such that:

lim ||u(t) - S(t)UI|Hs = 0. (1.12)

t-+koo

where S(t) denotes the free Schrddinger evolution operator. By unitarity of S(-) on

HS, it follows that (1.12) implies that |Iu(t)IIH is uniformly bounded. Several H8

'-scattering results have been shown NLS-type equations in [35, 37, 42, 43, 44, 75]. From H8 _{-scattering, one can deduce H}8_{-scattering if the initial data lies in H', for}

any s > si. This persistence of regularity result for scattering is sketched in Chapter

4.

Let us note that all of the mentioned scattering results hold on non-periodic domains. It is not expected to be possible to obtain scattering results on periodic domains due to weaker dispersion. This fact that L2 _{scattering doesn't hold was} precisely verified for the defocusing cubic NLS on T2 _{in [39].}

Another special situation occurs when the NLS equation is completely integrable. For our purposes, this means that there exist infinitely many conservation laws, which in turn give bounds on all Sobolev norms of degree a positive integer. More precisely, if (1.1) is completely integrable, k E N, and <D E H', then Iu(t)IIH is uniformly bounded in time. The most famous NLS equation which is completely is the cubic

NLS on R and on S1 T [84]. Another completely integrable model is the derivative

nonlinear Schr6dinger equation (DNLS), which is defined on R as the spatial domain with the nonlinearity K(u) = i6(|u|2_{u) [68]. We note that complete integrability}

doesn't immediately imply that

IIu(t)IIH

1.3.1

The linear Schr-dinger equation with potential

The growth of high Sobolev norms also been studied for the linear Schr6dinger equa-tion with potential, namely for a real funcequa-tion V = V(x, t), one considers the following linear PDE:

iut + Au = Vu. (1.13)

The growth of high Sobolev norms for (1.13) has been studied in [18, 17, 41, 111]. Under rather restrictive smoothness assumptions on V (for instance, in [18], V is taken to be jointly smooth in x and t with uniformly bounded partial derivatives with respect to both of the variables), it is shown that solutions to (1.13) satisfy for all e > 0 and all t E R:

||U(t)I|H s (1 ± t (1.14)

in [18], and, for some r > 0

|n(t)I|HS s,<k log(1 + tI)r. (1.15)

in [17, 111]. The latter result requires even stronger assumptions on V.

The idea of the proof of (1.14),(1.15) is to reduce the problem to one that is periodic in time and then to use localization of eigenfunctions of a certain linear differential operator together with separation properties of the eigenvalues of the Laplace operator on S1. These separation properties can be deduced by elementary

means on S'. In [18], the bound (1.14) is also proved on S', for d > 2. In this

case, the separation properties are proved by a more sophisticated number theoretic argument.

A different proof of (1.14) was later given in [41]. The argument given in [41] is

based on an iterative change of variable. In addition to recovering the result (1.14) on any d-dimensional torus, the same bound is proved for the linear Schr6dinger equation on any Zoll manifold, i.e. on any compact manifold whose geodesic flow is periodic. Moreover, in [110], it was shown that one can even obtain uniform bounds

on

||u(t)|Hs

6 < 1 is sufficiently small.

It would be an interesting project to obtain bounds of the type (1.14) for an NLS equation evolving from smooth initial data. Here, we have to restrict to an NLS equation for which H8-scattering is not known. Namely, as we noted above, H8

-scattering implies uniform bounds on |Iu(t)|IH.- Since one is considering initial data, one should also consider an NLS equation which is not completely integrable. Hence, a good model to consider would be the defocusing quintic NLS equation on S'. A possible approach to deduce (1.14) to substitute V =J12k into (1.13) and bootstrap polynomial bounds on |Iu(t)IIHS by applying the technique from [18] to obtain better bounds. However, there doesn't seem to be a simple way to implement this approach. The reason is that the reduction to the problem which is periodic in time doesn't work as soon as one has some growth in time of a fixed finite number of Sobolev norms.

The problem of Sobolev norm growth was also recently studied in [39], but in the sense of bounding the growth from below. In this paper, the authors exhibit the existence of smooth solutions of the cubic defocusing nonlinear Schr6dinger equation on T2 whose HS norm is arbitrarily small at time zero and is arbitrarily large at some large finite time. The work [39] is related to work of Kuksin [80] in which the author considers the case of small dispersion. By an appropriate rescaling, this can be shown to be equivalent to studying the same problem as in [39] with large initial data.

Furthermore, if one starts from a specific initial data containing only five frequen-cies, an analysis of which Fourier modes become excited has recently been studied in

[25] by different methods. One should note that both papers study the behavior of

the high Sobolev norms at a large finite time and that behavior at infinity is still an open problem.

Let us remark that in the mentioned works it is not clear if the constructed solution

u satisfies limsupt,±. |Iu(t)IIHs = o for s

>

to Bourgain [10, 11, 12]. In these papers, the KdV, NLS, and nonlinear Wave-type equation are studied respectively. However, one has to modify the original equations to look at a spectrally defined Laplacian or nonlinearity. The result in [12] gives a powerlike lower bound on the growth. The techniques are based on perturbation from the linear equations. It is not clear how to modify these methods to use them for the standard dispersive models. Furthermore, we note that if one considers a linear Schr5dinger equation with an appropriate random potential, the H' norm grows at least like a power of t almost surely [18].

A different way of modifying the NLS equation leads to the cubic Szeg6 equation:

iUt = 1l(Jul2u), x e S', E R (1.16)

u (x, 0) = <b (x) E L' (S').

Here, L2(S1) is the closed subspace of L2(S1) of functions having Fourier coefficients

with only non-negative indices, i.e. of the form ZkENo fkeikx and H1: L2(Si) -+ L (SI)

is the projection operator:

U(E aeikx) :

S

keZ kENo

The operator H is called the Szegd projection.

The analogous instability result to the one obtained in [39] for the equation (1.16) was recently obtained by Gerard and Grellier in [47] by using methods from complex analysis. It is also shown that (1.16) is completely integrable. On the other hand, there is no dispersive term in the equation, so the instability result is not unexpected.

1.4

Main ideas of our proofs

The main step in all of our proofs is to obtain a good iteration bound, based on an appropriate frequency decomposition. The iteration bounds we will use will usually not be as dependent on the structure of the nonlinearity as the iteration bound (1.9).

one with a smooth frequency cut-off, and one with a rough frequency cut-off. We always use the smooth frequency cut-off in the periodic setting, as the smoothness allows us to compensate for the lack of many dispersive estimates. In the non-periodic setting, we can use both types of frequency cut-offs, but in practice the rough cut-off is more useful only in the case of estimating the growth of fractional Sobolev norms of solutions to completely integrable equations such as the Cubic NLS on R.

1.4.1

The smooth cut-off; the upside-down I-method

We will use the idea, used in [18, 17, 111], of estimating the high-frequency part of the solution. Let El denote an operator which, after an appropriate rescaling, essentially adds the square L2 norm of the low frequency part and the square Hs norm of the high frequency part of a function. The threshold between the low and high frequencies is the parameter N > 1. With this definition, we want to show that there exist

#

depending on the nonlinearity and spatial domain and J, C > 0 depending only on <b such that for all times to:

C

El(u(to + 3)) < (1 + N )El(u(to)). (1.17)

One observes that (1.17) is more similar to (1.7) than to (1.9). The key fact to observe is that, due to the present decay factor, iteration of (1.17) O(N-) times doesn't cause exponential growth in E1_{(u(t)), as it did for ||u(t)||Hs in (1.8). We note}

that it is more difficult to obtain the decay factor in the periodic setting, than in the non-periodic setting.

We take:

Elf = |f|L2. (1.18)

Here D is an appropriate Fourier multiplier. In this paper, we take the D-operator to be an upside-down I-operator, corresponding to high regularities. We construct D in such a way that:

The operator D is the opposite from the standard I-operator, which was first developed in the work of Colliander-Keel-Staffilani-Takaoka-Tao (I-Team) [31, 32, 33, 34, 36].The idea of using an upside-down I-operator first appeared in [33], but in the low regularity context. The purpose of such an operator is to control the evolution of a Sobolev

norm which is higher than the norm associated to a particular conserved quantity. We then want to estimate:

Dut )|2dt. (1.20)

over an appropriate time interval I whose length depends only on the initial data. Similarly as in the papers by the I-Team, the multiplier 0 corresponding to the operator D is not a rough cut-off. Hence, in frequency regimes where certain cancela-tion occurs, we can symmetrize the expression and see how the cancelacancela-tion manifests itself in terms of 0, as in [33]. If there is no cancelation in the symmetrized expres-sion, we need to look at the spacetime Fourier transform. Arguing as in [22, 118], we decompose our solution into components whose spacetime Fourier transform is localized in the parabolic region (r + I |2) - L. In each of the cases, we obtain a

satisfactory decay factor. The mentioned symmetrizations and localizations allow us to compensate for the absence of an improved Strichartz estimate when working in the periodic setting. The localization to parabolic regions is particularly useful in the case of quintic and higher order NLS on S'.

In certain cases, we can add a multilinear correction to the quantity E1(u(t)), as

defined in (1.18) to obtain a quantity E2_{(u(t)) which is equivalent to E'(u(t)), but is} even more slowly varying, i.e. for which 3 in (1.17) is even larger. The idea is to choose the correction to be such that

jE

making E2(u(t)) even more slowly varying. Heuristically, we can view this is a way of

artificially adding more dispersion to the problem. This method is called the method

of higher modified energies, and was previously in the context of a modification of the

principle is reminiscent of the method of Birkhoff normal forms [4, 112] used in KAM theory. We apply the mentioned method in one and two-dimensional problems, both in the periodic and in the non-periodic setting. The two-dimensional setting is more subtle due to orthogonality issues that arise in studying the resonant frequencies.

1.4.2

The rough cut-off

We again start with a threshold N between the low and the high frequencies. Here, we take the rough projection Q defined by:

Qf()

The main idea of the method is to look at the high and low-frequency part of the solution u similarly as in [18], and, in addition, to use the bound on the integral Sobolev norms that one obtains from the complete integrability. Namely, for k E N:

||U(t)||Hk < B ((D (1.22)

From (1.22), we can deduce that for all times t:

|(I - Q)u(t)IIH. < C(<h)No. (1.23)

where a := s - [s] E [0, 1) is the fractional part of s. We note that the power of N is then in [0, 1) and is not s, as in (1.19). We use the estimate (1.23) to bound the low-frequency part of the solution.

The key is then bound

IIQu(t)I|HS.

#

equation, an increment 6 > 0, and C > 0, both depending only on the initial data

such that for all to c R, one has:

||Qu(to + 6)||2 < ( N )IQu(to)|12. + B1. (1.24)

for times to = 0, 6,..., n, where n E N is an integer such that n < N,- and to

telescope to obtain bounds on | Qu(t)

|Ha-This approach has been used to give bounds on the growth of fractional Sobolev norms for the defocusing cubic NLS on R. We have been able to derive only on R and not on S'. Our proof relies heavily on the fact that we are working on a non-periodic domain since we have to use improved bilinear Strichartz estimates, which are known not to hold in the periodic setting.

1.5

Some notation and conventions

We denote by A < B an estimate of the form A < CB, for some C > 0. on d, we write A <d B. We also write the latter condition as C = C(d).

number r, we denote by r+ the number r

+

<

r- is defined analogously as r - E. For 1 < p < oo, we define:

If C depends Given a real The number ||f||ILP(X) :=(if(x)|Pdx)P. JX and we write:

|f |ILoo(X)

(x)|.

Furthermore, given 1

<

|1W||L qLr(X xR) 111 ', i)h L(X) q I L[R)

If q = _{r, we observe that this is the norm}

||

|ILq,(XxR)-

||

||

|ILqLr

<

<

Hdlder conjugate exponent 1 < p' < oo by the formula: 1 + - = 1.

The Fourier transform

Given X = Rd or X - Td , and f

C

by:

f() := (1.25)

When X - Rd, the Fourier transform is defined on Rd, and when X - Td , it is

defined on Zd. In this case, we will usually denote the ( by n. (-, .) is defined to be

the L2_{-inner product on Rd, and}

Zd, when X = Rd and X = Td respectively. A key fact is Plancherel's Theorem:

IfIL2 ~ IfI||2

Given w E L2(X x R), we also define the spacetime Fourier transform by:

IX

_JR

(1.26)

(1.27)

Sobolev spaces

Let us take the following convention for the Japanese bracket

(-)

() := V1 + |x|2

.

Let us recall that we are working in Sobolev Spaces H* = H"(X) on the the domain

X, whose norms are defined for s E R by:

If IIH

=

(J

where f : X -+ C. Let us define H'(X) := (~),>Q Hs(X).

f

u(x, t)e -it -Edidx.

Free Schr6dinger propagator

We let S(t) denote the free Schr6dinger propagator. Namely, given

4

solution to:

it + Au

=0,

_{x E X, t R}

(1.30)

u(x, 0) = O(X).

is denoted by u(x, t) = (S(t)4)(x). By using the Fourier transform in x, one can

check that:

(S(t)4)(O) = e-"G (0). (1.31)

From Plancherel's Theorem (1.26), it follows that S(t) acts unitarily on L2_-based

Sobolev spaces.

XS'b spaces

An important tool in our work will also be Xsb spaces. These spaces come from the norm defined for s, b c R:

||U|IX.,b := (J(o 2s(_ + I22i( ~ T) 2drd()2. (1.32)

where u : X x R -+ C. The X5' spaces can be defined for general dispersive equations and are sometimes also called Dispersive Sobolev spaces. These spaces were first used in their present form in the work of Bourgain [9]. A similar type of space was previously used in the study of the one-dimensional wave equation by Beals [5] and Rauch-Reed [90]. Implicitly, X3,' spaces also appeared in the context of spacetime estimates for null-forms in the work of Klainerman and Machedon [78].

The Xsb spaces obey the structure of the linear Schr6dinger equation. If S(t) de-notes the linear Schr6dinger propagator as above, then one can check that (S(t)4)~((, T) is supported on the paraboloid r+

|I2

||II|X,b

are working on the spacetime domain X x R.

Littlewood-Paley decomposition

Given a function v E L2(X x R), and a dyadic integer N, we define the function

VN as the function obtained from v by restricting its spacetime Fourier Transform

to the region

|(|

Littlewood-Paley decomposition. In particular, we can write each function as a sum

of such dyadically localized components:

V~

VN-dyadic N

Multilinear expressions

We give some useful notation for multilinear expressions, which was first used in [31]. Let us first explain the notation when X - Td.

For k > 2, an even integer, we define the hyperplane:

k := f(ni,... ,nk) C (Zd)k : ni - - - -+ n = 0},

endowed with the measure 6(ni

+

+nk).

on 1

k, i.e. a k-multiplier, one defines the k-linear functional Ak(Mk; fi,..., fk) by: k

Ak(Mk;

fi

As in [31], we adopt the notation:

Ak(Mk; f) := Ak(Mk;f,f,. .

,

f,f).

When X - Rd, we analogously define:

1Fk := 61 .... , 4k) E (R2 )k : (1 -+ -. - - - ( = 0}.

In this case, the measure on 1

Pk is induced from Lebesgue measure d(₁ ... d$k_1 on

(R2

)k-1 by pushing forward under the map:

(6 1,...,( 4_1) 4 (6 1,...,( 1,41 -- -- - - - -01)

1.6

General facts from harmonic analysis

Strichartz estimates in the non-periodic setting

As was mentioned above, a fundamental tool we will have to use will be the Strichartz

estimates

Theorem 1.6.1. (Strichartz estimates for the Schr5dinger equation) We consider the domain Rd, and we say that a pair (q, r) is admissible if 2 < q, r oo, (q, r, d) z (2, oo, 2), and if the following relation is satisfied: 2 + = . If (q,r) and (q1,r 1) are

admissible exponents, then the following homogeneous Strichartz estimate holds:

||SMt)#|L qLr(RdxR) r5dAq,r 101ILL (Rd)- -1.3)

In addition, one has the inhomogeneous Strichartz estimate:

|| S(t - o-)F(o-)do-|jLLr(RdXR) 0d,q,r,qi,r |F. (1.35)

The non-endpoint case, i.e. when q, qi = 2 was first proved in [53, 114] and was based on the work of Strichartz [101]. The latter, in turn, was motivated by ear-lier harmonic analysis results in [93, 108]. The endpoint case q, qi = 2 was resolved in [69]. The reason why the endpoint case is so difficult is that the endpoint ver-sion of the Hardy-Littlewood-Sobolev inequality that one would like to use doesn't

hold. The authors of [69] get around this difficulty by using an appropriate dyadic decomposition.

In the mentioned papers, the key to prove Theorem 1.6.1 is to use the dispersive

estimate:

1

||S(t)4||1Lo(Rdy II-- IILI(Rd)- (1-36)

t 2

and combine it with an appropriate TT* argument. The bound (1.36) is shown as a consequence of the convolution representation of S(t)4 and Young's inequality.

Strichartz estimates in the periodic setting

Strichartz estimates are more difficult to prove on compact domains due to weaker dispersion. We observe that on a compact domain, the dispersive estimate (1.36) can't hold, since we can't have decay of the L' norm and conservation of the L2 norm. The local-in-time periodic analogue of (1.34), which is:

||S(t)4||

|' | |kIL2(

(1.37)

is known not to hold. In [9], it is shown that, when d 1 and N E N, one has:

N

||.e" IL(Tx[0,1]) > (log N) N .

n=1

Hence (1.37) can't hold in general. However, some positive results are known. They either require q to be smaller than 2(d+2) _d or that the function

4

fre-quency. In the latter case, one obtains a loss of derivative on the right-hand side of the inequality. More precisely, the bounds that one could expect are:

2(d + 2)

d di2 2(d +2) 1

||SMt)||L q:' [o')

<

N

when q

'

C

The first work dealing with these questions was that of Bourgain [9] in which (1.38) was proved in the special case q = 4, and (1.39) was proved in the cases d = 1, and

d = 2. When d = 3, (1.39) was proved under the additional assumption that q > 4.

Appropriate global-in-time versions were later proved in [58]. In both works, the key tool was to use lattice point counting techniques related to the work of Bombieri and Pila [8]. Let us note that some partial results on Strichartz estimates on the irrational torus have been proved in [20, 27]

Link between X5,' spaces and the Schr6dinger equation

The Xs,b spaces are well suited to the Schradinger equation. Let us briefly explain how one can see the connection. All of the facts we will mention now hold equally on Rd and on Td. They were already used in [9] and other works which first used Xsb space methods.

Given a Schwartz cut-off function in time q C S(R), the following localization

estimate holds:

||(t)S(t)#|xs,b

<

fIHs.

The following useful fact links Xsb spaces and Strichartz estimates:

Proposition 1.6.2. (c.f. Lemma 2.9 from [106]) Suppose that Y is a Banach space

of functions with the property that:

||e*07-S(t)$ljy < |||||H

As a consequence, we can deduce that for all pairs (q, r) for which the Strichartz estimate in the

||

||L

1

||U||L qL IXUIIXoa , whenever b> (1.41)

Improved bilinear Strichartz estimates

Strichartz estimates, and in particular the estimate (1.41) allow us to deduce mul-tilinear estimates by using H6lder's inequality. For example, if we consider u, v E

L2X(R x R), we can deduce that:

||UV||L

However, if we have further assumptions on the support of the Fourier transform of u(-, t) and v(-, t) in the space variable, it is possible to deduce an improved estimate. This key observation was first made by Bourgain in [14]. Later, a simplified proof was given in [31]. The improved bilinear estimate is:

Proposition 1.6.3. (Improved bilinear Strichartz estimate in the non-periodic

set-ting) Suppose N1, N2 > 0, with N1

>>

f,

such that for all t E R:

supp

f

y(t)

N2}-Then, the following bound holds: d-1

11fg1L <N2 2 (.3

||fg||L,(RdxR)

In particular, if d 1, we obtain a decay factor of - which is an improvement over the bound in (1.42).

Let us remark that the analogue of the Improved bilinear Strichartz estimate with a decay factor doesn't hold in the periodic setting. The following result is also due to Bourgain [9], in the case d = 2.

Proposition 1.6.4. (Improved bilinear Strichartz estimate for free evolution in the

periodic setting) Suppose N1, N2 > 0, with N1

>

f,g

Xoi F+qa x R) are such that for all t G R:

supp

f(t) C

{In

C

{|nl

Suppose that I C R is a compact time-interval. Then, given the following bound holds:

d-1 N 2

IX(t)fgIIL2 (72XI) _||~~g|jT'x)

<

IINIfIo,

As a consequence of Proposition 1.6.4, the following bound follows:

Proposition 1.6.5. (Improved bilinear Strichartz estimate in the periodic setting)

Suppose N1, N2 > 0, with N1

>

f,

that for all t G R:

supp

f

Suppose that I C R is a compact time-interval. Then, the following bound holds:

||fg||L2,(T2

$

f1

(1.45)

The idea to prove Proposition 1.6.5 from Proposition 1.6.4 is to use the Fourier inversion formula to write:

U(X, t) ~I e' S (t).F(S(- t) u)(x , r) d

and similarly for v and then use the Cauchy-Schwarz inequality in the parabolic variable together with the assumption that b = !+ > 1. These ideas are explained

in more detail in [22, 106]. We note that, when N1

>

can't be less than 1. A comprehensive survey about bilinear improved Strichartz estimates on the torus can be found in [48, 100]. We note that improved Bilinear Strichartz estimates have recently been studied in the case of compact Riemannian manifolds in [22, 23, 59].

In our proofs, we will use the bilinear improved Strichartz estimate to obtain a decay factor in the iteration bound. From the preceding discussion, we note that this estimate will be useful to this end only when we are working in the non-periodic setting. As a result, we will obtain better bounds on non-periodic domains. This is consistent with the heuristic that dispersion is stronger in the non-periodic setting.

1.7

Organization of the Chapters

In Chapter 2, we study the problem on S'. Here, we consider the defocusing power-type NLS and the Hartree equation, as well as other modifications of the defocusing cubic NLS. In Chapter 3, we study the problem on R. In this chapter, we find bounds on the growth of fractional Sobolev norms of solutions the defocusing cubic

NLS. In addition to the cubic NLS, we also consider the Hartree equation. Chapter

4 is devoted to the study of the problem on two-dimensional domains. We consider the problem both on T2 _{and on R}2_{. Results from Chapters 2 through 4 will be}

published in [97, 96, 95]. In Chapter 5, we study the Gross-Pitaevskii equation for dipolar quantum gases on R3_{, which is the physically the most relevant case. The}

results from Chapter 5 are the first step in a joint work with Kay Kirkpatrick and Gigliola Staffilani in which we plan to study the Gross-Pitaevskii equation for dipolar quantum gases in more detail [77].

Chapter 2

Bounds on

S

1

2.1

Introduction

In this chapter, we first study the 1D defocusing periodic nonlinear Schr5dinger equa-tion. Namely, given k E N and s

E

iUt

+

Au

=

ku, x

E S

_,t

E R

(2.1)

u(x, 0) = <D(x) c Hs(S').

The mass and energy are given by:

M(u(t)) := Iu(x,t)|2

_dX

_(Mass).

_(2.2)

and

E(u(t)) := |Vu(x, t)I2_{dx + 2 2}

j1u(x,

2k+2 (Energy). (2.3)

As was noted in [46, 84], the equation (2.1) is completely integrable when k = 1. Hence, if we start from smooth initial data, all the Sobolev norms of a solution will be uniformly bounded in time. We consider several modifications of the cubic NLS in which we break the complete integrability. The first modification we consider is the Hartree equation on Sl:

iUt

+

Au

= (V

*U

_)u,

_x

E S',t

_E

R

(2.4)

The assumptions that we have on V are:

(i) V E L1(S').

(ii) V > 0. (iii) V is even.

We can also break the integrability by adding an external potential on the right-hand side of the equation to obtain:

=|uU

₊

Au,

_x

_{G S,t E R}

(2.5) 0) = 4(x) C HS(Sl).

Here, we are assuming:

(i) A E co (Si).-(ii) A is real-valued.

Finally, we can add an inhomogeneity factor A into the nonlinearity, and obtain:

iUt

+

Au=

AU

_{u, x}

_E

_S

_{, t}

_E

_R

u(x, 0) = <>(x) E Hs(Sl).

(2.6)

Here, the inhomogeneity A = A(x) satisfies:

(i) A E C (S) (ii) A > 0.

2.1.1

Statement of the main results

The results that we prove are:

Theorem 2.1.1. Let k > 2 be an integer and let s > 1 be a real number. Let u be

a global solution to (2.1). Then, there exists a continuous function C, depending on

(s, k, E(<b), M(<b)) such that, for all t E R:

||u(t)||H-

<

M(<b))(1

It|)

For the modifications of the cubic NLS, we can prove the following results:

Theorem 2.1.2. Let s > 1 and let u be a global solution of (2.4). Then, there exists

a function C as above, such that for all t E R:

||u(t)I|Ha

<

tI)sI'II

Furthermore, we prove:

Theorem 2.1.3. Let s > 1 and let u be a global solution of (2.5). Then, there exists

a function C as above, such that for all t E R :

|Iu(t)IIH-

C(1

Itl)s+IKIH8.

(2.9)

Theorem 2.1.4. Let s > 1 and let u be a global solution of (2.6). Then, there exists

a function C as above, such that for all t C6 R :

|Iu(t)IIH.

It|)

_s+||4<b|H

8

.

(2.10)

It makes sense to consider the case k = 1 in Theorem 2.1.1, as long as we are taking s which is not an integer, and if we are assuming only 4 E H"(S1_{). It turns} out that we can get a better bound, which is the same as the one obtained for (2.4):

Corollary 2.1.5. Let s > 1 be a real number and let u be a global solution of (2.4).